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This research investigates the relationship between Feigenbaum fixed points and golden-mean scaling in Siegel disks.

July 29, 2026Open Access

Feigenbaum Fixed Point Hyperbolicity and Golden-Mean Scaling in Siegel Disks — E8 Intelligence Research

Key Points

This research investigates the relationship between Feigenbaum fixed points and golden-mean scaling in Siegel disks.
Utilized mathematical analysis of fixed points and scaling factors in quasiperiodic systems.
Examined the relationship between the golden ratio and rotation numbers in critical circle maps.
Established connections to renormalization group theory and parabolic domains.
Feigenbaum fixed point demonstrated hyperbolicity with δ ≈ 4.669 and α ≈ 2.502.
Golden ratio φ acting as the critical scaling factor linked to quasiperiodic dynamics.
Optimal convergence properties of the golden-mean rotation number ω = (√5-1)/2 highlighted in the analysis.

Abstract

FINDING: Feigenbaum renormalization fixed point exhibits hyperbolicity and universality for golden-mean Siegel disks, with golden ratio as the critical scaling factor in quasiperiodic dynamics. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; its reciprocal φ⁻¹ ≈ 0.618; Feigenbaum fixed point δ ≈ 4.6692016 (period-doubling), α ≈ 2.5029079 (scale factor); golden-mean rotation number ω = (√5-1)/2 ≈ 0.618; renormalization operator R acting on function space; Siegel disk boundary scaling by φ. | CONNECTION: Golden ratio φ and its reciprocal 0.618 appear as the universal rotation number for critical circle maps; the Feigenbaum tower's inflexibility links to parabolic domains with rotation numbers related to φ; the golden mean's continued fraction 1;1,1,1,... yields optimal convergence and is the "most irrational" number, central to KAM theory and quasiperiodic order. | DEPTH: 9 — This establishes a rigorous bridge between renormalization group theory (Feigenbaum universality) and quasiperiodic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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